Abstract: Let be a compact metrizable space and let be the Banach space of all real continuous functions defined on with the maximum norm. It is known that fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when contains a perfect set. However the space , where and is the first infinite ordinal number, enjoys the w-FPP, and so also satisfies this property if . It is unknown if has the w-FPP when is a scattered set such that . In this paper we prove that certain subspaces of , with , satisfy the w-FPP. To prove this result we introduce the notion of -almost weak orthogonality and we prove that an -almost weakly orthogonal closed subspace of enjoys the w-FPP. We show an example of an -almost weakly orthogonal subspace of which is not contained in for any .